Method of measuring out-of-roundness of machined parts



July 18, 1950 M. GOLDBERG 2,515,214

METHOD OF MEASURING OUT-OF-ROUNDNESS g OF MACHINED PARTS F iled June 21,1946 4 Sheets-Sheet l llllllllllllllllll INVENTOR MICHAEL GOLDBERG H BY(2 Mal W ATTORNEY July 18, 1950 M, GOLDBERG 2,515,214

METHOD OF MEASURING OUT-OF-ROUNDNESS 0F MACHINED PARTS Filed June 21,1946 4 Sheets-Sheet 2 [lllllllllll I Y- AXIS I l MEAN omega TANGENT TOCURVE CONTACT POINT OF TANGENT CURVE TO MEAN CIRCLE 0 CONTACT POINT OFTANGENT T0 CURVE a X-AXIS CENTER OF MEAN CIRCLE ECGENTRIGITY INVENTOR LENGTH OF NORMAL To TANGENT MICHAEL GOLDBERG ATTORNEY July 18, 1950 MGOLDBERG 2,515,214

METHOD OF MEASURING OUT-OF-ROUNDNESS OF MACHINED PARTS Filed June 21,1946 4 Sheets-Sheet 3 I80 DEGREES INVENTOR MICHAEL GOLDBERG ALL-ATTORNEY Flcrr. 7--

July 18, 1950 M. GOLDBERG 2,515,214

METHOD OF MEASURING OUT-OF-ROUNDNESS 0F MACHINED PARTS Filed June 21,1946 4 Sheets-Sheet 4 FFE B- INVENTOR MICHAEL GOLDBERG BY I v ATTORNEYPatented July 18, 1950 .METHOD .OFMEASIJRING OUT OF-ROUND- ?NESSOFiMAOHINED'fPARTS Michael-Goldberg, Washington, D. Application June.21,1946, Serial-No. 678,185

-1 Glaim. (Cl. "33*1378) (Granted under the act of .March 3, .1883, as

amended April 30, 1928; 3.7.0 0. 161.757)

'L'Ihe roundness -.of tubes, bars, a un rbarrels, cones mipes .andiother structures having external "or internal surfaces .of circularcross-section is {usually measured :by relatively rotating the structure:and .a :caliper. diameter I as measured .by the caliper is taken as ameasure of the eccentricity or out-of-roundnessof ;the.surface. Such.measurements are of :prime importance ;in :the testing of submarinehull, torpedoes, gun tubes, pipes, and other structure that may .besubjected to fluid pressures eitherifrom the outside or from the inside.Forexample, .thelhullofa submarine when measuredbyithe conventionalcaliper me'thodmay iniiicate perfect circularity, yet it may be .provedby more accurate methods or even by visualinspection that the surface is"not at all round. Any departure-from perfect circularity will not "onlyimpairithe e'ificiency of "the vessel'but will also subject it tounnecessary and tremendous stresses that may in some casesactually-crush the. hull.

'iEc'centricities in structures that are designed to be circular are duein most instances to lirregularities inmethods of manufacture. An 0bviou"source of error;.is animproper adjustment in a centerless grinder.Other causes .are .chattering cutting tools and'loose or worn'bearingsina lathe. Siniilarlmthe shape-of a bar before turningmay'lead 'itore'ccentricity. Thus, :if a bar of square .;cross+sectionzis:roug'h-turnedzto ;a 0:3 1- inder, the deflectionofith'e :cutting' toolwhen cutting the corners is greaterthanwhen cuttingthe sides and thiislikely to produce high spots at the corners. improper handling. Othersare warped by internal stresses in castings, forgings, 'and"heattreated,metals,;andbyaging and evaporation in plastics.

.If a cross-section is taken of a seemingly round structure, it will, ofcourse, define a closed curve. Many such curves are so developed thatwhen positioned between two parallel -,lines representing theplane'parallel faces of .aline caliper (or the two points of a :pointcaliper) they will be found to be rotatable between these parallel lineswithout at any time losing contact with both parallel-lines. S0 far athe conventional caliper methodnf .measuring eccentricity is concerned,.it ZIIISXOHIY necessary and sufiicient that ,the rotating curvemaintain .contact .with the *twopoin'ts or faces of "the caliper, in.:order to indicate circularity. "Sinceit can be demonstratedmathematically that there are many such curves that satisfy these'conditions and yet are The -.variation lot \the .Some materials aredeformed by I not circular, .it "will .he-evident that the conventionalmethodof testingeccentricity is defective. Consequently, :a caliper withtwo points .or .with plane parallel faces can not the relied onto-givevalid results.

The point-to-point .caliper .-is, .of .course, the

equivalent -.of an instrument with plane parallel "faces, so far asdetecting eccentricity is icon- .cerned. ,Moying :a lpointeto-pointexternal realiper back and forth ,around the .work gives the sameresults as rotating the .work .between "the plane-faces-of a.cali,perwith flat jaws. As .al ready indicated, csuchiresults .are not reliable.The present invention yconcerned -=primarily .with :the .design of .aninstrument ;for detecting or measuring eccentricity. For this ,purposaitis contemplated that the caliper .-f,or measurin .eaternal surf aceshave one jaw .or face; so eformed that .it ,is angular. ZI -he anglerisiso {chosen tha the device will --.be able -.to detect certain typesof eccentricity. If -.desired, the complementary straight vface {may beeither .opposed to the an.- .gular detecting face-or it,ma y been thesame side, .as .willr-bemade .amparent more ;particularly ,hereinafter.

The conventional point-to -point -internal caliper that is "used ,forinternal .surfaces is .defectime, :for the same reasons. This inventionalso provides .means for rectifying the terror and for detectingeccentricity -.in .inter-nal curves.

The invention will be-described in greater idetail with the aid .of theaccompanying drawings,

.which form part of this specification.

In :the drawings,

Fig. ,-1 shows .one form of non--circu1ar :eonvex cunve :that (may #38irotated between the two pa-rallel ;plane -.-f aces "of a l conventional:caliper while maintaining-contactwwith:thesegfacestatzallatimes.

Fig. .2 shows the same :icurve in a caliper alway- Iing a wedge :anglethat *passes ;the :particular curve as circular.

Fig. .13 :shows :a Zcaliper having ;;a wed e tangle of the (proper valueto detect :zeccentricities, [in accordance -with rthegprinciples of {theinvention.

.Fig. A :shows I a :caliper .giving :the same results as that of Fig.13,:but :having :a wedge 'facezthat .-is mountable ;en a -.c onvent;ional :caliper with plane parallelifaces.

.Fig..-.5:=.shows zanothericaliper for detecting ee- -.ce1tricities, buthaving :both zanyils can the :same :31 B.

:Fig. 6 :is .a graph illustrating ;a mathematical .analysisof thegprinciples .aof ;the ;inv.ention.

.Fig. -,-'7 is :a {graph indicating rhow :the ac-aliner wedge ran le ischosen.

Fig. 8 is a caliper constructed in accordance with the principles of theinvention, for measuring internal diameters.

Fig. 9 is another form of the instrument shown in Fig. 8.

It can .be shown mathematically that within every regular polygon anon-circular curve can be inscribed touching each and every side of thepolygon. Among all such curves that can be so inscribed in a givenregular polygon, there is one extreme case that is characterized bysharp corners. All the other inscribable curves for that particularpolygon may be derived as parallel curves of this extreme curve. I

A parallel curve to a given curve is the locus of points on the normalsto the given curve at a constant distance from the given curve.

All inscribed curves may be divided into two classes: namely, thoseinscribable in a regular polygon of an even number of sides, and thoseinscribable in a regular polygon of odd sides. As used herein, the termsodd and even refer to the number of sides ofthe regular polygon in whichthecurve is inscribed.

The eccentricity of curves inscribable in regular polygons of anoddnurnber of sides will be readily detected by'the-usual caliper withflat faces. The reason for this is that the number of bulges of theinscribed curve, being even where the number of sides of the regulapolygon is odd,

cannot be rotated between two parallel faces. Accordingly, the kind ofcurve above described that can be inscribed in a regular triangle,pentagon, or similar odd numbered polygon can be readily checkedfor'roundness by the ordinary flat face caliper. v a

On the otherhand, the curves inscribable within squares, hexagonsfoctagons, and other even numbered regulanpolygons maybe passed as roundby such a caliper, even when in fact they are not round, because due tothe odd number of bulges of the curve, there will always be a high pointcorresponding to alow' point, so that the curve may be rotatable withthe same two parallel faces without at any time losing contact 'With thetwo parallel faces.

The primary object of this invention is to devise some method ofdetecting the even polygon curves, since these will pass unnoticedwithin the fiat faces of a conventional caliper. Angular faces may beused, some more effectively than others. Some angle are particularlyunsuitable since they too will pass certain curves. These unsuitableangles are the angles'between the alternate sides of a regular polygonof an even number of sides, for they will pass non-circular curves thatwould pass by the flat angle test also. Similarly, a wedge angle equalto the angle between any pair of sides separated by an odd number ofsides will not detect non-circularity of even polygon curves. a

The following is an analysis of a noncircular convex curve'that'may berotated within two or more fixed lines and yet remain in contact withall of these lines atfall times, with particular reference to Fig. 6 ofthe drawings.

Take a regular polygon of m sides and denote the magnitude of any one ofits external angles by the letter 0. It is required to construct such acurve C that it may rotate within the polygon and yet keep in contactwith all of the sides at all times. For such a curve it is .necessaryand sufiicient that the'normals at the points of contact of thecurvewith the side of the polygon be concurrent, that is, th'at 'theyintersect at a -point in the polygon to these sides.

point. This point represents the instantaneous center of rotation andwill, of course, vary as the curve is rotated. It follows from planegeometry that since the polygon is regular the sum of these normals willremain constant. The area of a regular polygon may be divided intotriangles whose bases are the sides "of the polygon and Whose altitudesare the perpendiculars from any Since this constant area is equal tohalf the sum of these altitudes multiplied by one of the bases, the sumof the altitudes must be constant. This sum will equal m times theradius of an inscribed circle of the polygon. This circle, of course,will be fixed depending on the size and the number of sides of thepolygon and will be independent of the position of the instantaneouscenter of rotation.

Construct the polygon about the center of rotation in such a way thatone of the sides will make the angle with the horizontal axis of thesystem of co-ordinates. The desired curve is in scribed within thispolygonJ -Let one side of the polygon coincide with a tangent T'to thecurve so that the tangent makesthesameangle with the axis. 1 Then, sincethe polygon is regular, the sum of the normals is constant and is equalto m times a, where a is the radiusof the inscribed circle. The equationrepresenting the sum may be set forth as Equation 1. 1 J 4ma=p()+p(+0)+p(+20)+ +p(+(m-l)' 0) (1) In this equation, p denotesthedistance of the initial side of the polygon from theorigin of 00-ordinates. It should be noted that this distance is taken fromthe'origin to the tangent T of the curve (or the side of the polygon),and that it does not represent thedistance from the origin to the pointof contact of the curve with'the polygon.

Instead of keeping the polygon fixed and revolving the curve within itto maintain contact with all sides at all times, we may consider thecurve as fixed. If the surrounding regular poly-'- gon is rotated so asto keep in contact with the curve within it, Equation l may berepresented in polar tangential form as Equation 2 wherewis any angle. v,V v Because the polygon is regular, 0, the external angle, equals andconsequentlywe have Equationfi v cos+cos(+0) +cos(+'20)+'-' Y '+cos'(+(m1)a.) =0 (5) F() satisfies the condition set forthin Equa tion 3 if weevaluateit-as j r, I

resonant where 7 denotes ;=an y arbitrary zone-valued :fllnoitdnniwhatsoeyer. substituting in Equation :2 we have Equation "7 whichrepresents the polar tangential equation \9 aicurye that may :be:rotated within a regular ;;polygpn1of.-m sides while it maintainscontact with .all the sides .at all times.

lt-canbe sho-wnthat the normals to the sides .01 the ,polygon at thepoint-of contact are con- .cm ent forsuch a curve, .for Equation 8ifulfilled-foranyvalue at all of o. --Take Equation -9 v =b cos Margawhere his a positive constant. Then, substituting in Equation '7, wehave 1 =11 1) cos -cos (10.)

p'(q5)= b sin 45 cos g'd-gb cos sin m b (11) 2 %b cos cos T04: 12 i'I'heCartesian loo-ordinates of the point of contact of the ,side T of theregular polygon making an angle with the initial .X-axis are, ingeneral, Equations 13 and 14 see ".-D,iifer,entialgleichungenLosungenmethoden eund Liisungen, E. Kamke, 3rd edition (1944), p. 1 1.

Again it is to be noted that the point of congtfiCtrOf the-curveC w iththeside'To-f the polygon isidiflerent from the perpendicular normalpoint.

Substituting Equation 1 in Equation 13 we have Equations 15 and 1.6

x=a sin g1: cos sin out (15) 6 :synnnetrical with :respect :to theY-axis and to the straight :line

Lbrd

The extreme case Where the curve has sharp corners is obtained by takingb equal to and therefore may be taken as a, measure of the eccentricityof the sharp cornered curve. In consequence, as m increases,this-measure of non-circularity decreases rapidly. In other words, .asthe number of sides of the polygon increases, the in- .scribed curverapidly approaches the form of a circle.

As 'hereinbefore indicated, the primary object of this invention is todevise a, caliper that can detect noneci-rcularity of a closed convexcurve that can be inscribed and rotated within a regular polygon of aneven number of sides. A caliper having one fiat-face and one angularface is used for this purpose, and it is required to ascertain the angleof the angular face so that it will be most efficacious in detectingnon-circularity of the curves hereinbefore mentioned. Insuch a caliper,one base of the polygon may be assumed to be in contact with the fiatface of .the caliper, and the curve is positioned so that the polygon.is symmetric with respect to the angular face of the caliper. It can beshown that there are no non-circular closed curves which can rotate in atriangle and keep contact with the three sides. of the triangle at eachorientation unless the sides of the triangle lie on the sides of someregular polygon. Therefore, if one wishes to make an effective gageemploying three contacts for detecting or measuring eccentricity(zout-of-roundness), the choice of these contacting faces must be madesothat these faces do not-lie on the faces of regular polygon.

:In Fig. 1 isshown a conventional caliper having a flat face H and asecond flat face I2 paral- ;lelther eto. Suppose it is required todetect noncircnlarityof a curve [3.

If this curve l3 were .inscribable in a regular polygon of odd sides,the caliper of Fig. '1 could be eificacious for the desired purpose, ashereinbefore described, because the curve could not rotate within thecaliper. If however, the curve I3 is inscribable in a regular polygon Mof an even number of sides, such as the hexagon shown, it will beobvious, in View of the preceding analysis, that the curve l3 may berotated, or conversely, that the caliper may be 7 rotated about thecurve 1'3, without at any time non-circular curves.

losing contact between the curve l3"'and the two faces H and 12. In sucha case, the caliper will not detect non-circularity of the curve.

In Fig. 2 is shown the same curve l3. The caliper, however, has anangular face 15 in place of the straight face I2 of Fig. 1. Let it beassumed that the angle is 60 degrees. 'It will be apparent that thepolygon within which the curve may be rotated concurs with the Wedgeangle of 60 degrees, since the alternate sides of the hexagon make thisangle with each other. Consequently, the curve It may be rotated withinthe caliper of Fig. 2 and maintain contact with both sides of theangular face I as well as with the fiat face I I, even though in fact itis not circular. As a result, this caliper, too, will be unreliable forchecking surfaces. Indeed, since the maximum eccentricity of the curveI3 is In other words, if the wedge angle of the face I5 is 60 degrees,convex surfaces having an eccentricity of .055 may pass as round.Conversely, a caliper face l5 having an angle of 60 degrees, as well asa face I2 of 180 degrees is unsuitable for detecting eccentricity if thepolygon within which the curve is inscribable has six sides.

The deviation angle, i. e., the exterior angle between the extension ofone side and its next adjacent side of a regular polygon of m sides is0=21r/m. Therefore, the angle between a pair of alternate sides is1r-41r/m. Ingeneral, the angle between a pair of sides separated by anodd number of other sides is 1r4K1r/m=1r(1-4K/m) where K is an integer.

Therefore, the poorest wedge angles for use in an eccentricity gage isgiven by 1r(14K/m) where K/m is a proper fraction in its lowest termsand m is a small even number. For example, when 112:6, and K=l, then1r(l4K/m)=1r/3=60f. In this case there are curves of eccentricity 2/m/3s= /1a which will pass as round. For m=8, and K=1, then1r(14K/m)=1r/2=90.

In this case there are curves of eccentricity 2/m /64 /32 which willpass as round.

A chart or graph can be prepared to show the unsuitable angles forvarious inscribable curves. Such a graph is shown in Fig. "7. On thisgraph, the abscissa of each vertical line represents an unsuitable anglederived from a regular polygon of m sides while the ordinate, which is2/mis the extreme eccentricity of inscribable and rotatable curvesdescribed herein. Some angles are obtainable from different values of mand different values of K; therefore several values of eccentricityshown by the small black circles lie on the corresponding verticallines. It will be seen from this figure, for example, that both 60degrees and 90 degrees are unsuitable, The figure represents all suchunsuitable or bad angles for detecting curves inscribable in polygonshaving the listed number of sides, the number of sides m and the maximumeccentricity being shown on the Y-ordinate.

The problem resolves itself, therefore, into finding a wedge angle thatwill effectually detect Common angles like 30, 45, 60, 90, 120, and 180will all pass many non-circular curves as circular.

Even though the selected angle for the supporting wedge angle ofaneccentricity gage is 'tical purposes.

tice, the wedge angle will have certain manufacturing allowances ortolerances. Therefore, instead of a mathematically exact selected angle,one must consider the effectiveness of the totality of angles in a smallrange of angles. Since the possible eccentricities'of curves associatedwith'a polygonwill decrease as the square ofthe number ofsid'es of thepolygon, it is important to select an angle suchthat, over the rangeofth'e angles manufactured to the specification of a. given angle andits tolerances, none of the angles are exactly equal to the angle of aregular polylgon of a small number of sides.

Furthermore, an angle close to the angleof a regular polygon is notsensitive to eccentric curves belonging to that polygon. Therefore, itisde; sirable to select "angles as far as possible from poor angles. Asone moves away from one poor angle, one will approach another poorangle. It is desirable to choose the angle by weighing the eccentricityhazards of the nearby angles. For example, a satisfactory wedgeangle maybe selected from the chart which is thegapproximate mean of a range ofangles over which there are no vertical lines, 1. e., there are noangles within the foregoing range equal to 1r(1-4K/m) where m is aninteger equal to v e being the lowest value of eccentricity that willnot be detected by the angle expressed by 1r(14K/m) where K is anyintegral value less than m /4.

It is best to use the approximate mean of such a range because it willbe observed "froml' the chart that an angle of, say 59will not passl'anycurves in the range shown, that is up to m'=5 5. Theoretically an angleof 59 degrees would be excellent for detecting non-circular curves. Itshould be noted however, that this is so close to 60 degrees that it maypass many of the noncircular curves that are passed by the 60 degreeangle, unless'the operator is extremely careful and unless theinstrument is extremely sensitive. For these reasons, it is desirable tohave an angle somewhat removedfrom those that are definitely unsuitable.A preferred angle would be closer to 86 degrees. This is four degreesremovedfr'om a right angle and is enough to distinguish there'- from andis not too close to the next vertical line although its lowereccentricity is favored. A wedge angle of 86 degrees will pass, as notedin Fig. '7, a curve inscribable in some regular polygon of more than 50sides when such a curve is not completely circular, but the maximumeccentricity of such a curve will be only This amounts to 0.0008 form=50. Such eccentricity is so low as to be negligible for most prac- Ineffect, therefore, such a curve may be called round. A wedge anglehaving an error of one degree on the low side of the angle of 86 degreeswill pass a curve inscribable'in -a polygon having more than l5 sides,which would also involve an error so low'as to be practicallynegligible. 7

By the same token, an an'glej'of 94 degreesjwill be highly suitable fordetecting non-circular curves. This, too, is sufiiciently removed from90 degrees to avoid error due on that account, so that any curves thatwill be passed by a 95 degree caliper will be substantially round.

What is required therefore, is a caliper having one fiat face II and anangular face l6 opposing the flat face II, the angle being so chosen asto detect curves that are in fact non-circular up to a certain degree.This angle may be close to 86 degrees or 94 degrees or some othersimilarly chosen angle. The diameter that is to be measured may be readdirectly on a suitable scale H.

In practice, a caliper with parallel faces II and I2 will first be usedto detect non-circular curves that may be inscribed in polygons havingan odd number of sides, as already indicated. Those curves that arepassed as apparently round by the parallel faced caliper, will then beinserted in a caliper having the angular face l6, and such caliper willdetect eccentricities that would otherwise remain undetected.

If desired, a wedge face I 8 may be machined to fit over the flat faceI2 so that only one caliper need be used.

As shown in Fig. 5, the flat face usually opposing the angular face l6may comprise a member IS on the same side. Such a device will beparticularly useful for detecting or measuring large curves or surfaces.

Any of the calipers may have a gage or Vernier for measuring thedepartures from circularity.

For measuring bores internally, a device such as that shown in Fig. 8may be employed. This utilizes the same principles. The two arms 2| and22 are fixed relatively to each other at the desired angle of 86 degreesor 94 degrees or other chosen value. The end of each arm has atelescopic contact 23 or 24, respectively. The instrument is adapted tobe inserted into the bore, as close to the center as possible. Thecontacts 23 and 24 are then slid along the arms 2| and 22 until theymake contact with the internal surface of the bore, and are then clampedinto position on the arms 2] and 22. A third arm 21, symmetricallypositioned with respect to the arms 2| and 22 in fixed relationshipthereto, has a slidable telescopic contact 28. The contact is free toslide on the arm 21 at all times. The device is rotated (or the bore isrotated), and the movement of the contact 28 may be read on the gage 29.Whereas an angle of 60 degrees between the arms 2! and 22 would beunsuitable, for the reasons already specified, and would pass as roundcurves that actually have an eccentricity of as much as 0.055, an angleof 86 degrees will readily detect an eccentricity as small as 0.0008.

An apparently round internal curve of this class is such that a regularpolygon of any given odd number of sides can be inscribed within it andmade to rotate within it so that all the vertices of the polygon retaincontact with the curve at all positions of the polygon. These curves arenot necessarily constant width curves that are inscribable and rotatablein a regular polygon. Such an internal curve may be obtained by tracingthe path of any vertex of a regular polygon of any given odd number ofsides, which polygon is made to rotate about a fixed inscribable curvethat keeps contact with all sides of the polygon. Since the polygonsdictate the critical angles, the angles that are unsuitable for theconstant width or external curves are similarly unsuitable for theinternal curves hereinbefore mentioned.

In another form of the internal caliper, shown in Fig. 9, the two arms3| and 32 may be fixed at half the suitable angle, such as 43 or 47degrees. The contacts 33 and 34 are moved along the arms 3| and 32 untilthey contactthe sides of the bore and are then fixed in position. Theapex of the angle has a movable contact 35 that is movable and incontact with the bore at all times during relative rotation if the boreand the caliper, and the movement of this contact 35 may be read on thegage 36. The device Will operate the same way as that shown in Fig. 8.

It will be seen, therefore, that a caliper constructed in accordancewith the principle of the invention will be highly useful in detectingor measuring eccentricities of either internal or external curves.

The invention described herein may b manufactured and used by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

I claim:

The method of measuring out-of-roundness of a machined part to detectany eccentricity (e) in said part comprising plotting points onCartesian coordinates of a series of angles (4) corresponding to each ofa plurality of values of eccentricity (e), said plurality of values ofeccentricity (e) being expressible by the equation e:2/m wherein mincludes all integral values up to the computed m for the selectedeccentricity (c), said series of angles for each value of eccentricitybe ing obtained from the equation 4:1r(14K/m wherein K takes allintegral values from 1 up to m/4, selecting from said plot theapproximate mean of the range of angles between any pair of immediatelyconsecutive points, cutting a notch having an angular valueapproximately equal to said selected value, placing said part in saidnotch, bringing an indicator into contact with said part, and noting thedeviations of the indicator during rotation of said part in said notch.

MICHAEL GOLDBERG.

REFERENCES CITED The following references are of record in the file ofthis patent:

UNITED STATES PATENTS

